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Keywords:
one-side porosity; local strong upper porosity; completely strongly porous set; ideal
Summary:
Let $\rm SP$ be the set of upper strongly porous at $0$ subsets of $\mathbb R^{+}$ and let $\hat I(\rm SP)$ be the intersection of maximal ideals $\boldsymbol {I}\subseteq \rm SP$. Some characteristic properties of sets $E\in \hat I(\rm SP)$ are obtained. We also find a characteristic property of the intersection of all maximal ideals contained in a given set which is closed under subsets. It is shown that the ideal generated by the so-called completely strongly porous at $0$ subsets of $\mathbb R^{+}$ is a proper subideal of $\hat I(\rm SP).$ Earlier, completely strongly porous sets and some of their properties were studied in the paper V. Bilet, O. Dovgoshey (2013/2014).
References:
[1] Bilet, V. V., Dovgoshey, O. A.: Investigations of strong right upper porosity at a point. Real Anal. Exch. 39 (2013/14), 175-206. MR 3261905
[2] Chinčin, A.: Recherches sur la structure des fonctions mesurables. Russian, in French Moscou, Rec. Math. 31 (1923), 265-285, 377-433.
[3] Denjoy, A.: Leçons sur le calcul des coefficients d'une série trigonométrique. Tome II. Métrique et topologie d'ensembles parfaits et de fonctions. French Gauthier-Villars, Paris (1941). Zbl 0063.01081
[4] Denjoy, A.: Sur une propriété de séries trigonométriques. French Amst. Ak. Versl. 29 (1920), 628-639.
[5] Dolženko, E. P.: Boundary properties of arbitrary functions. Math. USSR (1968), 1-12 translation from Izv. Akad. Nauk SSSR Ser. Mat. 31 (1967), 3-14 Russian. MR 0217297
[6] Dovgoshey, O., Riihentaus, J.: Mean value type inequalities for quasinearly subharmonic functions. Glasg. Math. J. 55 (2013), 349-368. DOI 10.1017/S0017089512000602 | MR 3040867 | Zbl 1271.31008
[7] Foran, J., Humke, P. D.: Some set-theoretic properties of $\sigma$-porous sets. Real Anal. Exch. 6 (1980/81), 114-119. MR 0606545
[8] Humke, P. D., Vessey, T.: Another note on $\sigma$-porous sets. Real Anal. Exch. 8 (1982/83), 262-271. MR 0694514
[9] Karp, L., Kilpeläinen, T., Petrosyan, A., Shahgholian, H.: On the porosity of free boundaries in degenerate variational inequalities. J. Differ. Equations 164 (2000), 110-117. DOI 10.1006/jdeq.1999.3754 | MR 1761419 | Zbl 0956.35054
[10] Kechris, A. S.: Hereditary properties of the class of closed sets of ubiqueness for trigonometric series. Isr. J. Math. 73 (1991), 189-198. DOI 10.1007/BF02772948 | MR 1135211
[11] Kechris, A. S., Louveau, A., Woodin, W. H.: The structure of $\sigma$-ideals of compact sets. Trans. Am. Math. Soc. 301 (1987), 263-288. MR 0879573 | Zbl 0633.03043
[12] Przytycki, F., Rohde, S.: Porosity of Collet-Eckmann Julia sets. Fundam. Math. 155 (1998), 189-199. MR 1606527 | Zbl 0908.58054
[13] Repický, M.: Porous sets and additivity of Lebesgue measure. Real Anal. Exch. 15 (1989/90), 282-298. MR 1042544
[14] Semenova, O. L., Florinskii, A. A.: Ideals of porous sets in the real line and in metrizable topological spaces. J. Math. Sci., New York 102 (2000), 4508-4522 translation from Probl. Mat. Anal. Russian 20 (2000), 221-242. DOI 10.1007/BF02672903 | MR 1807069
[15] Thomson, B. S.: Real Functions. Lecture Notes in Mathematics 1170 Springer, Berlin (1985). MR 0818744 | Zbl 0581.26001
[16] Tkadlec, J.: Constructions of some non-$\sigma$-porous sets on the real line. Real Anal. Exch. 9 (1983/84), 473-482. MR 0766073
[17] Väisälä, J.: Porous sets and quasisymmetric maps. Trans. Am. Math. Soc. 299 (1987), 525-533. DOI 10.2307/2000511 | MR 0869219 | Zbl 0617.30025
[18] Zajíček, L.: On $\sigma$-porous sets in abstract spaces. Abstr. Appl. Anal. 2005 (2005), 509-534. DOI 10.1155/AAA.2005.509 | MR 2201041
[19] Zajíček, L.: Porosity and $\sigma$-porosity. Real Anal. Exch. 13 (1987/88), 314-350. MR 0943561 | Zbl 0666.26003
[20] Zajíček, L.: On cluster sets of arbitrary functions. Fundam. Math. 83 (1973/74), 197-217. MR 0338294
[21] Zajíček, L., Zelený, M.: On the complexity of some $\sigma$-ideals of $\sigma$-$P$-porous sets. Commentat. Math. Univ. Carol. 44 (2003), 531-554. MR 2025819 | Zbl 1099.54029
[22] Zelený, M., Pelant, J.: The structure of the $\sigma$-ideal of $\sigma$-porous sets. Commentat. Math. Univ. Carol. 45 (2004), 37-72. MR 2076859 | Zbl 1101.28001
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